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Chapter 5: Problem 71

At the Fermi National Accelerator Laboratory (Fermilab), a large particle accelerator, protons are made to travel in a circular orbit \(6.3 \mathrm{~km}\) in circumference at a speed of nearly \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\). What is the centripetal acceleration on one of the protons?

Short Answer

Expert verified
The centripetal acceleration of the proton in the Fermilab's particle accelerator is \(9.0 \times 10^{13} \, \text{m/s}^2\).

Step by step solution

01

Calculate the Radius

Calculate the radius (\(r\)) of the circular path using the formula \(r = \frac{C}{2\pi}\), where \(C = 6.3 \times 10^{3} \,\text{m}\). Conversion into meters is necessary to match the units of speed.
02

Substitute Values in the Radius Formula

Substitute \(C = 6.3 \times 10^{3} \,\text{m}\) into the formula to solve for \(r\).\n\[r = \frac{6.3 \times 10^{3} \,\text{m}}{2\pi} = 1000 \,\text{m}\]
03

Calculate Centripetal Acceleration

Now substitute the values of \(v = 3.0 \times 10^{8} \,\text{m/s}\) (provided speed) and \(r = 1000 \,\text{m}\) (calculated radius) into the formula for centripetal acceleration \(a = \frac{v^2}{r}\).
04

Substitute Values in the Acceleration Formula

Substitute \(v = 3.0 \times 10^{8} \,\text{m/s}\) and \(r = 1000 \,\text{m}\) into the formula to solve for \(a\).\n\[a = \frac{(3.0 \times 10^{8})^2}{1000} = 9.0 \times 10^{13} \, \text{m/s}^2\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Centripetal Acceleration
Centripetal acceleration is an essential component of circular motion. It refers to the acceleration required to keep an object moving in a circular path. Without it, the object would move in a straight line due to inertia. The centripetal acceleration always points towards the center of the circle. This direction is essential for maintaining the circular path of the object.

To calculate centripetal acceleration, the formula \( a = \frac{v^2}{r} \) is used, where \( v \) is the velocity of the object and \( r \) is the radius of the circle. The quadratic relation shows that doubling the velocity results in four times the acceleration for the same radius, highlighting the strong impact of velocity on centripetal acceleration.

The significance of centripetal acceleration can be seen in particle accelerators like Fermilab, where maintaining a precise path is crucial for particles traveling at nearly the speed of light.
Particle Accelerator Physics
Particle accelerators are powerful tools in physics and technology. They speed up particles, such as protons, to extremely high velocities. This is done primarily to study subatomic particles and the fundamental forces of nature. The circular shape is vital as it allows particles to be accelerated continuously using magnetic fields.

At Fermilab, protons circulate through a 6.3 km ring at velocities close to the speed of light. This setup requires precise control of centripetal forces to ensure these particles do not deviate from their intended path. Magnetic fields provide the necessary force that acts as centripetal force. As the acceleration of these particles is immense, new physics phenomena can be observed upon collision with other particles.

Particle accelerator physics has numerous applications, including medical uses like cancer treatment, materials science for creating new materials, and even in archeology to analyze historical artifacts without damaging them.
Kinematic Equations
Kinematic equations are mathematical relationships that describe the motion of objects. In the context of circular motion, these equations help determine quantities such as speed, velocity, and acceleration.
  • Velocity: Refers to the speed of the object moving along the circular path. For Fermilab, this velocity is extremely high, i.e., \(3.0 \times 10^{8}\, \text{m/s}\).
  • Radius: The distance from the center of the circle to the path followed by the particle. At Fermilab, this was shown to be 1000 meters.
  • Acceleration: Using the centripetal acceleration formula, one can solve for acceleration once velocity and radius are known.
The kinematic equations enable physicists to understand and predict the behavior of particles within accelerators, helping in planning experiments and analyzing results efficiently. Mastering these concepts is crucial for anyone delving into advanced physics, especially fields dealing with high-energy particles and quantum mechanics.

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Most popular questions from this chapter

Occupants of cars hit from behind, even at low speed, often suffer serious neck injury from whiplash. During a low speed rear-end collision, a person's head suddenly pivots about the base of his neck through a \(60^{\circ}\) angle, a motion that lasts \(250 \mathrm{~ms}\). The distance from the base of the neck to the center of the head is typically about \(20 \mathrm{~cm}\), and the head normally comprises about \(6 \%\) of body weight. We can model the motion of the head as having uniform speed over the course of its pivot. Compute your answers to the following questions to two significant figures. (a) What is the acceleration of the head during the collision? (b) What force (in newtons and in pounds) does the neck exert on the head of a \(75-\mathrm{kg}\) person in the collision? (As a first approximation, neglect the force of gravity on the head.) (c) Would headrests mounted to the backs of the car seats help protect against whiplash? Why?

A 7.6-kg object rests on a level floor with a coefficient of static friction of \(0.55\). What minimum horizontal force will cause the object to start sliding? SSM

You want to push a heavy box of books across a rough floor. You know that the maximum value of the coefficient of static friction \(\left(\mu_{\mathrm{s}}\right)\) is larger than the maximum value of the coefficient of kinetic friction \(\left(\mu_{\mathrm{k}}\right)\). Should you push the box for a short distance, rest, push the box another short distance, and then repeat the process until the box is where you want it, or will it be easier to keep pushing the box across the floor once you get it moving?

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